Irreducibility of a general polynomial in a finite field

  1. 1. The problem statement, all variables and given/known data

    For prime p, nonzero [itex]a \in \bold{F}_p[/itex], prove that [itex]q(x) = x^p - x + a[/itex] is irreducible over [itex]\bold{F}_p[/itex].


    2. Relevant equations



    3. The attempt at a solution

    It's pretty clear that none of the elements of [itex]\bold{F}_p[/itex] are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if [itex] \alpha[/itex] is a root of q(x), then so is [itex]\alpha + 1[/itex], and from there I was able to deduce (hopefully not incorrectly) that [itex]\bold{F}_{p}(\alpha)[/itex] is the splitting field for q(x) over the field, but I can't figure out the degree of the extension, so I'm kind of stuck. Any ideas?
     
  2. jcsd
  3. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    How about showing that if q(x) is reducible, then it must split into linear factors?
     
  4. I sat and thought about it for about 45 minutes and played around on a chalkboard with it, and I'm not sure how to go about your suggestion. Is it related to the work I already did, or is it a completely new line of attack? Thanks for helping.
     
  5. Okay, I figured it out. Thanks a lot.
     
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